Topics are arranged as per SYLLABI published by Maharashtra State Board of Secondary and Higher Secondary Education.

_{1} and m_{2} separated by a distance *r*, that is represented by the relationship .

Where G is the universal gravitational constant. The proportionality constant G is called universal Gravitational constant. In SI System, value of gravitational constant G is 6.672 X 10^{-11} Nm^{2} Kg^{-2} Dimensional formula of G is M^{-1}L^{3}T^{-2}

% increase in g = 2 (% decrease in R) = 2 X 1% = 2 %.

Thus acceleration due to gravity on the earth’s surface would increase by 2%.

Let’s assume a satellite is launched into a circular orbit of radius R around earth. A second satellite is launched into an orbit of radius (1.01)R. The period of second satellite is larger than that of the first one by approximately 1.5%.

Time period of satellite .

Percentage change in time period = (% change in orbital radius) = (1) = 1.5

Law of Orbits | Law of Areas | Law of Periods |
---|---|---|

Every planet moves in an elliptical orbit around the sun, with sun situated at one of the foci of the ellipse. | The line joining a planet to the sun sweeps equal areas in equal intervals of time, howsoever small these time intervals may be. | The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet. |

Note: The Sun is not at the center of the ellipse, but is instead at one focus. | L = Angular Momentum of planet and m= mass of planet. | ; |

Note: The Earth-Sun distance is constantly changing as the planet goes around its orbit. | The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion. |
If distance between earth and the sum becomes half its present value, the number of days in a year will become 129. ; ; T_{2} = 365 / 2 √2 = 129 days. |

The minimum energy required to remove a satellite from the earth’s gravitational field to infinity is called binding energy of a satellite.

Escape velocity is the minimum velocity with which a body should be projected from a given surface so as to enable it to just overcome the gravitational pull of that surface (eg: earth). The value of escape velocity from the surface of a planet of mass M, radius R and acceleration due to gravity g is . Escape velocity does not depend upon the mass or shape or size of the body as well as the direction of projection of the body. For earth value of escape velocity is 11.2 Km s^{-1}. A planet will have atmosphere if molecular velocity of atmospheric gases are much less than the escape velocity. Artificial satellites are man made satellites launched from the earth. The path of these satellite are elliptical with the centre of earth at a foci of the ellipse. However, as a first approximation we may consider the orbit of satellite as circular.

Heavenly Body | Escape Velocity |
---|---|

Moon | 2.3 Km s^{-1} |

Mercury | 4.28 Km s^{-1} |

Earth | 11.2 Km s^{-1} |

Jupiter | 60 Km s^{-1} |

Sun | 618 Km s^{-1} |

Neutron Star | 2 X 10^{5} Km s^{-1} |

If the ratio of the radii of two planets is 'a' and the ratio of the acceleration due to gravity on their surface is 'b'; then ratio of the escape velocities is .

Weightlessness (or Zero gravity): State of a body such that the set of gravitational and inertial forces to which it is subject have a zero resultant.

You might have seen pictures of astronauts floating inside a space shuttle as it orbits Earth. The astronauts are said to be weightless, as if Earth’s gravity were no longer pulling on them. Yet the force of gravity on the shuttle is almost 90 percent as large as at Earthâ€™s surface.

Key Points to note about weightlessness condition in orbit:

- A falling object is in free fall if the only force acting on it is gravity.
- Weightlessness occurs in free fall when the weight of an object seems to be zero.
- Objects orbiting Earth appear to be weightless because they are in free fall in a curved path around Earth

The acceleration produced in the motion of a body under the effect of gravity (earth’s gravitation) is called acceleration duet to gravity (g). If we consider a body of mass m placed at the surface of the earth, then force of gravity acting on it is and acceleration due to gravity is Where = mean density of the earth. For the earth, mean value of g on the earth’s surface is 9.8 ms^{-2}

- Value of acceleration due to gravity at a height h from the surface of the earth is given by

If h R then or - Value of acceleration due to gravity g at a depth d from the surface of the earth is given by or At the centre of the earth d=R and hence g
^{’}=0 - As earth rotates about its own axis with a period of 24 h (or angular velocity ), all bodies situated on the earth’s surface move along the circular path and need centripetal force for that. A part of weight of body is used to provide centripetal force and consequently effective value of g changes to

Where = Latitude angle of given place

At poles = 90^{o}and hence g_{pole}= g

At equator = 0^{o}and hence

A communication or geosynchronous or geostationary satellite is a satellite in geosynchronous orbit with an orbital period the same as the earth’s rotation period. Such a satellite returns to the same position in the sky after each sidereal day, and over the course of a day traces out a path in the sky. A special case of geosynchronous satellite is the geostationary satellite, which has a geostationary orbit - a circular geosynchronous orbit directly above earth’s equator.

- INSAT-3D is one of the communication satellite of India.
- The orbit of the satellite must be circular and in the equatorial planes of the earth.
- The angular velocity of the satellite must be in the same direction as the angular velocity of rotation of the earth.
- The period of revolution or the satellite must be equal to the period of rotation of earth about its axis.

Earth revolves around the sun in circular path and required centripetal force is provided by gravitational force between earth and sun , but the work done by this centripetal force is zero.

Escape Velocity = X Orbital Velocity

If orbital path of satellite is circular , then its speed is constant and if the orbital path of satellite is elliptical, then its speed in its orbit is not constant. In that case aerial velocity is constant.

When rocket is launched from west to east in equatorial plane, the maximum linear velocity is added to the lunching velocity of the rocket, due to which launching becomes easier.

Total energy of earth (planet) satellite system is independent of eccentricity of orbit and it depends on semi-major axis and masses of the planet and satellite.

As spacecraft moves away from the surface of earth towards moon, then there will be no change in the mass of spacecraft. However its weight will keep on changing as described below:

- It’s weight will decreases in the beginning.
- It will become zero at some point, where the force of attraction on the spacecraft due to earth and that due to the moon become just equal and opposite.
- It will again start increasing as the spacecraft further moves towards the moon.

- Gravitational forces between two bodies form an action-reaction pair. i.e. they are equal in magnitude and opposite in direction.
- Gravitational force between two bodies is independent of the presence or absence of other bodies.
- Gravitational force is always attractive.
- According to Newton, the word gravity refers to the force exerted on objects by the earth. It holds the atmosphere around the earth.
- Orbital velocity does not depend on the mass of satellite. It depends on the mass & radius of the planet around which the rotation is taking place and the radius of the orbit. Its direction at an instant is along the tangent to path at that instant.
- Greater the height of the satellite smaller is the orbital speed and vice versa.
- If a satellite is revolving around a planet in an elliptical orbit, its orbital speed is not uniform.
- Work done by a satellite in a complete orbit is zero.
- The centripetal acceleration of the satellite is equal to the acceleration due to gravity.
- P.E. of a satellite orbiting in a circular orbit is always more than its K.E. If height of the satellite is increased then it’s P.E. increased while K.E. decreases.